Risk-Neutral Probabilities, No-Arbitrage Pricing, and Market Completeness
Summary
The document explains the link between absence of arbitrage and risk-neutral probability measures in derivative pricing. Under the fundamental theorem of asset pricing, an arbitrage-free market model admits a risk-neutral measure, subject to the theorem’s model assumptions. Prices can then be expressed as discounted expectations of future payoffs under that measure. With a constant interest rate, the risk-neutral expected future share price corresponds to the no-arbitrage forward price; a call option is valued by discounting its risk-neutral expected payoff.
It also distinguishes existence from uniqueness. A risk-neutral measure is unique when the market is complete, meaning that payoffs can be replicated by self-financing trading strategies. The document notes that proving the converse direction of the theorem can require substantial mathematical conditions; in practice, model builders often construct a measure directly, for example using a change-of-measure technique. These are general pricing principles, and their use depends on the assumptions and completeness of the chosen market model.
Key ideas
- An arbitrage-free model admits a risk-neutral probability measure under the theorem’s assumptions.
- Derivative prices can be represented as discounted risk-neutral expectations of their payoffs.
- The risk-neutral expected share price gives the no-arbitrage forward price with a constant rate.
- Completeness implies uniqueness of the risk-neutral measure, while incomplete markets can have multiple measures.
- The existence and equivalence results depend on the mathematical conditions of the model.
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# How are the two concepts No arbitrage & Risk neutral probability related?
# How are the two concepts No arbitrage & Risk neutral probability related?
The title, and might I add, that this question is in relation to the Black-Scholes model and why the concepts are important for option pricing in general.
## Answer by febstar (score 5)
https://quant.stackexchange.com/a/37509
A market model is arbitrage-free if and only if it has a risk-neutral probability measure. This is the fundamental theorem of asset pricing.
That is, in a securities model, the two concepts are one and the same. You can think of the risk-neutral probabilities as those that give the arbitrage free prices of derivatives.
Suppose the interest rate, $r$, is constant. Then, for example, the time $T$ expected value of a share, $S_t$, where the expectation is taken using risk-neutral probabilities, is $S_t e^{r(T-t)}$, which you might recognise as the no-arbitrage forward price.
In a similar way, the no-arbitrage value of a call option at time 0 is the discounted expectation (using risk-neutral probabilities) of the call payoff, i.e., $$C = e^{-rT} \mathbb{E_Q}[\max(S_T - K,0)].$$
This method can be used to obtain the no-arbitrage price of derivatives in general. You'd want this price to ensure that you do not expose yourself to any losses from being arbitraged.
## Answer by Antoine Conze (score 3)
https://quant.stackexchange.com/a/37510
Absence of arbitrage is in general considered equivalent to the existence of a risk neutral probability measure.
The measure is unique iif the market is complete (meaning any payoff can be replicated with a self financing hedging portfolio).
While it is easy to prove that the existence of a risk neutral measure implies absence of arbitrage, the reverse is more complicated to show. In the discrete case it makes use of the Hahn Banach theorem, and in the continuous case it requires quite a bit of technical conditions, so when working on a particular model one simply builds the risk neutral measure to show its existence, usually using the Girsanov theorem.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.